Summary: The general linear model: what you need to know
P.M.E.Altham, Statistical Laboratory, University of Cambridge.
October 14, 2004
Note: this set of basic notes contains deliberate gaps, for you to ll in.
The models
y i = + x i + i ;
y i = + x i +
x i
2 + i
and
y ij = + i + j + ij
may all be seen as special cases of the model
y i = T x i + i
for i = 1; :::; n where we assume that ( i ; i = 1; :::; n) form a random sample
from N(0; 2 ). Here y i is the `dependent' variable, x i is the known covariate,
the unknown parameter, of dimension say p, and i is the unknown `error': we
assume that ( i ; i = 1; :::; n) NID(0; 2 ), ie ( i ; i = 1; :::; n) form a random
sample from N(0; 2 ). The parameter 2 is also unknown. We rewrite this
model as
Y = X + ;
where Nn (0; 2 I) (a multivariate- normal distribution).